What if you had a polynomial expression like
in which some of the terms shared a common factor but not all of them? How could you factor this expression? After completing this Concept, you'll be able to factor polynomials like this one by grouping.

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Guidance

Sometimes, we can factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called
factor by grouping.

The next example illustrates how this process works.

Example A

Factor
.

Solution

There is no factor common to all the terms. However, the first two terms have a common factor of 2 and the last two terms have a common factor of
. Factor 2 from the first two terms and factor
from the last two terms:

Now we notice that the binomial
is common to both terms. We factor the common binomial and get:

Example B

Factor
.

Solution

We factor 3x from the first two terms and factor 4 from the last two terms:

Now factor
from both terms:
.

Now the polynomial is factored completely.

Factor Quadratic Trinomials Where a ≠ 1

Factoring by grouping is a very useful method for factoring quadratic trinomials of the form
, where
.

A quadratic like this doesn’t factor as
, so it’s not as simple as looking for two numbers that multiply to
and add up to
. Instead, we also have to take into account the coefficient in the first term.

To factor a quadratic polynomial where
, we follow these steps:

We find the product
.

We look for two numbers that multiply to
and add up to
.

We rewrite the middle term using the two numbers we just found.

We factor the expression by grouping.

Let’s apply this method to the following examples.

Example C

Factor the following quadratic trinomials by grouping.

a)

b)

Solution:

Let’s follow the steps outlined above:

a)

Step 1:

Step 2:
The number 12 can be written as a product of two numbers in any of these ways:

Step 3:
Re-write the middle term:
, so the problem becomes:

Step 4:
Factor an
from the first two terms and a 2 from the last two terms:

Now factor the common binomial
:

To check if this is correct we multiply
:

The solution checks out.

b)

Step 1:

Step 2:
The number 24 can be written as a product of two numbers in any of these ways:

Step 3:
Re-write the middle term:
, so the problem becomes:

Step 4:
Factor by grouping: factor a
from the first two terms and a -4 from the last two terms: